Self-assembly of colloid-cholesteric compositesprovides a possible route to switchable optical

materials

K. Stratford∗, O. Henrich∗, J. S. Lintuvuori∗,M. E. Cates, D. Marenduzzo†

SUPA, School of Physics and Astronomy, The King’s Buildings,The University of Edinburgh, Edinburgh, EH9 3JZ, UK

∗ equally contributed to this work

† Corresponding author: dmarendu@ph.ed.ac.uk

Colloidal particles dispersed in liquid crystals can form new ma-terials with tunable elastic and electro-optic properties. In a pe-riodic ‘blue phase’ host, particles should template into colloidalcrystals with potential uses in photonics, metamaterials, and trans-formational optics. Here we show by computer simulation that col-loid/cholesteric mixtures can give rise to regular crystals, glasses,percolating gels, isolated clusters, twisted rings and undulatingcolloidal ropes. This structure can be tuned via particle concen-tration, and by varying the surface interactions of the cholesterichost with both the particles and confining walls. Many of thesenew materials are metastable: two or more structures can ariseunder identical thermodynamic conditions. The observed struc-ture depends not only on the formulation protocol, but also on thehistory of an applied electric field. This new class of soft materialsshould thus be relevant to design of switchable, multistable devicesfor optical technologies such as smart glass and e-paper.

When spherical colloidal particles are mixed into a nematic liquid crystal,they disrupt the orientational order in the fluid and create defects (disclina-tion lines) in the nematic close to their surfaces. To minimise the free energy

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cost, it is generally advantageous for defects to be shared: therefore the col-loidal particles have a generic tendency to aggregate. Such aggregates mayfurther self-assemble into lines [1], 2D crystals [2], planar structures [3], or3D amorphous glasses [4]. Besides being of fundamental interest to materialsscience, these structures may have tunable elastic and optical properties [5].Hence they offer exciting prospects for applications as biosensors [6], or asnew devices [7, 8].

But one may also choose to disperse colloidal particles in a cholesteric,rather than nematic, liquid crystal. The molecules making up this materialare chiral, and this causes their average orientation (described by a direc-tor field) to rotate in space in a helical fashion, rather than remain uniformas in nematics. This typically creates a 1-dimensional periodic structure,called the cholesteric phase, whose wavelength is the pitch, p. The periodic-ity can however become fully three dimensional in the so-called blue phases(BPs) [9]. BPs arise because the director field can twist around more thanone direction at once (see the cartoon in Supplementary Fig. 1). Such doubletwist regions (generally cylinders) are energetically favoured at high enoughmolecular chirality, but lead to geometric frustration because it is impossibleto tile the whole 3D space with double twist cylinders without also intro-ducing “disclination lines” (These are singular topological defects on whichthe director field is undefined.) In blue phases these disclination lines them-selves either form a 3D periodic regular lattice (in so called BPI and BPII),or remain disordered (BPIII) [10]. BPs are stabilised by the tendency ofchiral molecules to twist, but destabilised by the cost of forming topologicaldefects (which locally diminish the molecular alignment). In conventionalcholesteric materials [9] this balance is only achievable in a narrow temper-ature range of a few degrees, around the onset of liquid crystalline ordering.However, recent advances in formulation have widened this stability rangeenormously [11, 12, 13, 14]. Most recently, templated BPs with a stabilityrange of -125 to 125C have been reported [15], paving the way to applicationsof BPs in operational display devices.

Besides being remarkable materials in themselves, BPs offer significantpromise as hosts for dispersing colloidal particles. Because BPs contain adisclination network even in the absence of particles, they can potentiallytemplate colloidal self-assembly. This was proved recently by simulationsof BP-dispersed nanoparticles which exclude the surrounding liquid crystalbut otherwise have negligible surface interaction with it (to give a so-called‘weak anchoring’ regime) [16, 17]. Such particles are attracted to the discli-

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nation network in BPs: by covering up the defect cores, the high elasticenergy cost of those regions is avoided. The pre-existing order of the defectnetwork then templates particles into a regular colloidal crystal of the sameperiodicity. Because this structure has a wavelength in the visible range,the resulting material should inherit (and probably enhance) the incompletephotonic bandgap of the parent blue phase, as exploited for instance in laserdevices [12]; for related possibilities see [18].

A second compelling reason to study colloidal-BP composites is that, evenwithout particles, BPs can show several competing metastable free energyminima, corresponding to different topologies of the defect network [19, 20],with a strong dependence on applied electric or magnetic fields [21]. Addingparticles is likely to further enrich the free energy landscape, creating com-posites that might be promising candidates for bistable or multistable de-vices, in which energy is needed only to switch optical properties and notto maintain them. (This is the e-paper paradigm [22], and can lead to hugeenergy savings.) To explore such metastability, and switching strategies be-tween states, we require multi-unit-cell, time-dependent simulations. Thesego beyond previous studies of colloidal-BP composites [17] but follow compa-rable simulations of pure BPs [10, 21, 23], and isolated and dimeric colloidalparticles in cholesterics [24, 25].

The simulations presented here are representative of a liquid crystal bluephase with unit cell size in the range of 100–500 nm [9], and particles with adiameter of around 50 nm. The simulations address both the bulk structure,and the structure which forms when the colloidal dispersion is confined ina narrow sandwich of width comparable to the unit cell size [26]. It is seenthat a wide variety of structure is possible, and may be influenced by param-eters including the solid volume fraction, and the details of the anchoring atthe colloid surface (and the at the surface of the confining walls). Further,evidence is found for metastable switching of this structure in an appliedelectric field. Details of the computational approach are set out in the Meth-ods section, while simulation parameters are reported in the SupplementaryMethods.

ResultsStructure in bulk BPI. We show in Fig. 1 (a–d) the final state structuresof four simulations in which we first equilibrated a stable BPI disclinationnetwork in bulk (with periodic boundary conditions) and then dispersed col-loidal particles at random within it. Among dimensionless control parameters

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are the particle solid volume fraction φ (φ = 1% or 5%); the ratio r = R/λof particle radius to the BP lattice parameter λ (here r ' 0.14) and the ratiow = WR/K where W is the anchoring strength of the cholesteric at the parti-cle surface, and K is the elastic constant of the liquid crystal. The anchoringstrength is defined such that W > 0 imparts a preferred orientation to thedirector field at the particle surface. This orientation can be either normalto the surface or in a plane tangential to the surface, a property related tosurface chemistry in experimental systems. We choose w = 0.23, 2.3 whichare within typical experimental range [4]. This parameter can be viewedas the ratio of the anchoring energy, WR2, to the elastic energy scale fordistortion KR.

For a single colloidal particle in a nematic fluid, w controls the formationof a local topological defect at the particle surface [5], with weak distortionof the nearby fluid at small w. In the colloidal-BP system, w also plays adetermining role in the final composite structure. When w is small (Fig. 1aand Fig. 1b; Supplementary Movies 1 and 2), the attraction of particles tothe disclination lattice causes the lattice to become covered by particles, withlittle disruption of its long-range order. Thus a templated colloidal crystal iscreated, as first reported in [17]. Interestingly, the weak but finite anchoringapparently creates an elastic force between the colloidal particles which leadsto very slow dynamics (Supplementary Fig. 4) and the formation of smallcolloidal lines within disclinations, especially for dilute samples (Fig. 1a).It seems likely that polymer-stabilised blue phases [11] work by a similarprinciple to this small w case, with a weak segregation of monomers towardsthe defect regions.

For large w (Fig. 1c and Fig. 1d; Supplementary Movies 3 and 4), the pic-ture changes dramatically. Each colloidal particle now disrupts significantlythe order of the nearby fluid, with defect formation close to its surface [5].The resulting strong disruption of local liquid crystalline order around eachparticle leads now to a strong interparticle attraction. This restructures thedisclination network completely, destroying the long range order of the orig-inal BPI topology. Most notably, particle aggregation favours the formationof defect junctions, where four disclinations meet. This structural motif ispresent in the unit cell of BPII rather than BPI (and also seen in the amor-phous BPIII [10]). The particle clusters are disjoint at low volume fractions(φ ' 1%) but they interact elastically via the connecting disclinations. Atlarger φ, the aggregates join up to form a percolating colloidal cluster whichessentially templates the disclinations rather than vice versa.

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As colloidal-nematic composites [4], all the structures in Fig. 1 (a–d)should be soft solids, of nonzero elastic modulus G at low frequency. Thiscontrasts with pure BPs, where G = 0 as the fluid can flow with a finiteviscosity via permeation of molecules through a fixed defect lattice [27, 28].We expect the structures in Fig. 1 (a–c) to be only weak gels, as here thedefect network percolates but particle contacts do not. This is broadly similarto the cholesteric-nanoparticle gel reported in [29] (with G < 1 Pa). Incontrast, the colloidal gel structure in Fig. 1d should be much more resistantmechanically, due to the formation of a thick percolating particle network.The structure is now akin to that of the self-quenched glass found in colloidal-nematic composites [4] at much higher volume fraction (over 20% as opposedto below 5% here), for which G ∼ 103 Pa. This figure is consistent with anestimate obtained by comparing the mechanical properties of the networkof defects stabilised by colloidal particles with those of rubber [30] for therelevant simulation parameters (see Supplementary Table 1).

A more practical protocol for including particles in a BPI phase is to dis-perse the particles at random in an isotropic phase, and then quenching thisinto the temperature range where BPI is stable. Here we know that, withoutparticles, the kinetics can favour disordered (BPIII-like) intermediates whichmay be long lived or even metastable [23]. Fig. 1 (e–h) shows outcomes ofsuch a quench protocol with the same parameter sets as in Fig. 1 (a–d). Forsmall w, the particles again decorate the disclination lines, but are this timetemplated onto an amorphous network rather than an ordered one. Increas-ing the volume fraction (Fig. 1f) leads to a more even coverage with a betterdefined particle spacing. For large w, on the other hand, strong interpar-ticle forces again dictate the self-assembly. At low concentrations, discreteparticle clusters decorate an amorphous defect lattice (Fig. 1g) whereas athigher φ a percolating network of colloidal strands (the gel in Fig. 1h) isagain seen. The structures in Fig. 1 (a–h) have been followed for ∼20 ms(see Supplementary Fig. 4); they correspond to either very slowly evolvingstates or deep metastable minima (elastic interactions are of order KR anddwarf thermal motion, see Supplementary Fig. 4 and Ref. [5, 18]).

It may be possible to anneal the structures obtained via the quench route,although we have not attempted this in simulation. The results presentedsuggest that even when starting with colloidal particles randomly dispersedin an ordered blue phase, the particles are able to distort the structure(Fig. 1 a,b). This suggests that it would be difficult to identify a quenchprotocol which would lead to less, rather than more, disorder.

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Structures in confined BPI. In display devices and other optical appli-cations, it is typical for liquid crystals to be strongly confined (and subjectto strong boundary effects), in contrast to the bulk geometries consideredabove. We therefore next describe how colloidal suspensions self assemblein thin sandwiches (thickness h = 1.75λ), whose confining walls can favoureither tangential (planar degenerate) or normal (homeotropic) orientation ofthe director field. For simplicity we here impose one or the other as a strictboundary condition.

Fig. 2 (a–d) shows the case of normal wall anchoring on varying φ and w.We quench from the isotropic to the ordered phase as was done in Fig. 1 (e–h), and again find a metastable, amorphous disclination network (see Supple-mentary Methods). The resulting colloidal-cholesteric composites are some-what similar to those seen in the bulk: for small w the particles are tem-plated by the network (Fig. 2a,b), whereas for large w we observe isolatedclusters (Fig. 2c) and percolated colloidal gels (Fig. 2d), for φ = 1% and 4%respectively. There are, however, significant effects of the confining walls.First, the normal anchoring at the wall recruits particles there, giving (forthe chosen h) a density enhancement of the colloid particles at both walls(see side-views in Fig. 2a–d). Second, the clusters formed at large w arethemselves anisotropic with an oblate habit. These clusters are similar tothose reported experimentally for large colloid particles in a cholesteric (butnon-BP) host [31].

Fig. 2 (e–h) shows comparable simulations for planar wall anchoring.Without colloidal particles, planar anchoring results in a regular cholesterichelix, with axis perpendicular to the boundary walls. This twisted texturedevelops via a set of defect loops, which shrink and disappear to leave adefect-free sample. However, dispersing particles with small w arrests theloop shrinking process, stabilising twisted rings (Fig. 2e) at small φ, and morecomplicated double-stranded structures at larger φ (Fig. 2f). For larger w,we observe thick colloidal “ropes” about 5 particles across (Fig. 2g,h). Theseropes undulate to follow the chiral structure of the underlying fluid. On in-creasing the density they develop branch points and finally again approach apercolating gel morphology. Note that planar wall anchoring is incompatiblewith the normal anchoring on the colloidal particles (for w > 0), which arethus repelled from both walls (see the side views in Fig. 2e–h).

The degree of similarity between the bulk and confined structures is some-what masked by the local change in particle density in the confined case. Ifone allows for this, Figs. 1 and 2 show the individual structural motifs are

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actually rather similar: templated lines for weak anchoring and clusters forstrong anchoring.

Switching in electric field. An interesting question, relevant for applica-tions to devices, is whether these structures can be modified, or switched,with electric fields. Fig. 3 addresses this question, starting from amorphousblue phase III, this time at a higher chirality. Dispersing colloidal particlesin this network results in the particle occupying the available nodes, wherethe order parameter is low (Fig. 3a; Supplementary Movie 5). Applying ahigh enough field along the x direction leads to a switching to a new phase,where colloidal arrays fit within a network with distorted hexagonal cells(Fig. 3b and Supplementary Fig. 7; Supplementary Movie 6). Remarkably,upon field removal this honeycomb structure does not find its way back to theconfiguration before switching on the field, but gets stuck into a metastablestructure (Supplementary Fig. 7), with residual anisotropy along the fieldswitching direction. Cycling the field along the x and y directions leads todistinct metastable structures, which are retained after field removal (Sup-plementary Fig. 8 and Fig. 9). We note that for the fluid parameters usedhere, the field induced changes are irreversible with the weakly anchoringcolloid particles. The mesophase without colloid particles undergoes a re-versible ordering transition [10]; the reversibility is retained in the presenceof particles only if their anchoring is very weak.

DiscussionThe surprising wealth of colloidal-BP composites obtained in Figs. 1-3 is thecomplex consequence of a simple competition between two self-assembly prin-ciples. First, blue phases provide a three-dimensional template onto whichcolloidal particles are attracted so as to reduce the elastic stresses arisingat the disclination cores [17]. Second, colloidal intrusions in a liquid crystalraise the free energy locally; sharing this cost creates a generic tendency toaggregate into clusters [4]. Which of these factors dominates depends on w,a dimensionless measure of the anchoring strength of the director field atthe colloidal surface. For w 1, the templating principle dominates. Col-loidal crystals, gels or twisted rings can then arise. For w 1, interparticleattractions defeat the templating tendency, and we predict disconnected ag-gregates, percolating gels, or helical colloidal ropes. In both cases, controlis offered by the process route, colloidal concentration, and the anchoringconditions at sample walls. We have also shown that electric fields can beused to switch between metastable states, providing a possible route to future

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multistable device applications.How do these simulations relate to experiment? Several authors have used

nanoparticles in the context of stabilising BPs [32, 33]. These nanoparticlesare typically a few nanometres in size, in which case stabilisation is explainedby the mechanism of defect removal. Experiments suggest this mechanismis also relevant for larger particles in the range of 40–70 nm [34], which isconsistent with the current simulations. However, the mechanism is observedto be less efficient for particles above 100 nm, where particles may start todisrupt longer-range liquid crystal order [34]. While the field of self-assemblywith nanoparticles is relatively new [35], nanoparticles have also been usedin both nematic [36] and chiral liquid crystals [37] as host fluid templates.The variation of structure seen in our simulations suggest that some carefulcharacterisation of parameters such as surface anchoring of particles (whichare often stabilised, or even functionalised with surface ligands), togetherwith those associated with any confining surfaces or interfaces, is required tounderstand the final structure and to be able to reproduce it with a givenexperimental protocol. The identification of a model system which could al-low unambiguous comparison between experimental systems and simulationwould be extremely useful. Such a system might adopt particles of diame-ter in the range 50 nm or above, which allows a coarse-grained simulationapproach of the type used here to be adopted, which can then capture thelarge-scale structure required to compare with experiment. In the same vein,detailed surface chemistry is difficult to represent in coarse-grained simula-tions, so heavily functionalised particles might be avoided.

We have not explored in this work switchability by flow [38], but thiscould again be inherited from the underlying BP dynamics [19, 28], in whichcase the same materials may also be relevant to the emerging technology ofoptofluidics [39].

MethodsHere, the simulation method is described. Note that all quantities are

expressed in simulation units. Details of the simulation parameters, andtheir mapping to physical units, may be found in Supplementary Table 1and the Supplementary Methods.

Thermodynamics. The thermodynamics of the cholesteric liquid crystalcan be described by means of a Landau-de Gennes free energy functional F ,

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whose density is written as f :

F [Q] =∫d3rf(Q(r)). (1)

This free energy density f = f(Q) may be expanded in powers of the orderparameter Q and its gradients; Q is a traceless and symmetric tensor whichis denoted from now on in subscript notation as Qαβ. The largest eigenvalueq and corresponding eigenvector nα of Qαβ describe the local strength andmajor orientation axis of molecular order. The theory based on the tensorQαβ, rather than a theory based solely on the director field nα(r), allowstreatment of disclinations (defect lines) in whose cores nα is undefined.

Explicitly, the free energy density is:

f(Qαβ) = 12A0

(1− 1

3γ)Q2αβ − 1

3A0γQαβQβγQγα + 1

4A0γ(Q2

αβ)2

+ 12K(εαγδ∂γQδβ + 2q0Qαβ)2 + 1

2K(∂βQαβ)2. (2)

Here, repeated indices are summed over, while terms of the form Q2αβ should

be expanded to read QαβQαβ; εαγδ is the permutation tensor. The first threeterms are a bulk free energy density whose overall scale is set by A0 (discussedfurther below); γ is a control parameter related to a reduced temperature.Varying the latter in the absence of chiral terms (q0 = 0) gives an isotropic-nematic transition at γ = 2.7 with a mean-field spinodal instability at γ = 3.

The remaining two terms of the free energy density in Eq. 2 describedistortions of the order parameter field. In theoretical work and when de-scribing a generic rather than a specific system, it is conventional [9, 40] toassume that splay, bend and twist deformations of the director are equallycostly, that is, there is a single elastic constant K. The parameter q0 is re-lated to the helical pitch length p via q0 = 2π/p, describing one full turn ofthe director in the cholesteric phase.

There are two dimensionless numbers, which are commonly referred toas κ, the chirality, and τ , the reduced temperature [9] which can be usedto characterise the system. In terms of the parameters in the free energydensity, these are:

τ =27(1− γ/3)

γ(3)

κ =

√√√√108 K q20A0 γ

. (4)

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If the free energy density Eq. 2 is made dimensionless, τ appears as prefactorof the term quadratic in Qαβ, whereas κ quantifies the ratio between bulkand gradient free energy terms. The chirality and reduced temperature maybe used to characterise an equilibrium phase diagram of the blue phases inbulk cholesteric liquid crystals [9, 21].

The effect of an electric field can be modelled by including the followingterm in the bulk free energy density:

− εa12π

EαQαβEβ, (5)

where Eα are the components of the electric field, and εa (here assumedpositive) is the dielectric anisotropy of the liquid crystal. The strength ofthe electric field is quantified via one further dimensionless number

E2 =27εa

32πA0γEαEα. (6)

The quantity E is known as the reduced field strength.

Surface free energy. In addition to the fluid free energy density f(Qαβ),a surface free energy density is also present to represent the energetic costof anchoring at a solid surface. In the case of normal anchoring, the surfacefree energy density (per unit area) is

fs(Qαβ, Q0αβ) = 1

2W (Qαβ −Q0

αβ)2, (7)

where Q0αβ is the preferred order parameter tensor at the solid surface, and

W is a constant determining the strength of the anchoring. In the case ofplanar anchoring, a slightly more complicated expression is required to allowfor degeneracy (see, e.g., [44]). The determination of Q0

αβ for normal andplanar anchoring is discussed below. For a colloidal particle of radius R,the strength of the surface anchoring compared which the bulk fluid elasticconstant may be quantified by the dimensionless parameter WR/K. Forsmall values of this parameter, the presence of a particle surface should havelittle impact on the local fluid LC ordering.

Dynamics of the order parameter. A framework for the dynamics ofliquid crystals is provided by the Beris-Edwards model [41], in which thetime evolution of the tensor order parameter obeys

(∂t + uν∂ν)Qαβ − Sαβ = ΓHαβ. (8)

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In the absence of flow, Eq. 8 describes a relaxation towards equilibrium ona timescale determined by a collective rotational diffusion constant Γ . Thisrelaxation is driven by the molecular field Hαβ, which is the functional deriva-tive of the free energy with respect to the order parameter [41]:

Hαβ = − δFδQαβ

+ 13δαβTr

(δFδQαβ

). (9)

The tensor Sαβ in Eq. 8 completes the material derivative for rod-likemolecules [41]. It couples the order parameter to the symmetric and anti-symmetric parts of the velocity gradient tensor Wαβ ≡ ∂βuα. The symmetricpart Aαβ and the antisymmetric part Ωαβ are defined as

Aαβ = 12(Wαβ +Wβα), (10)

andΩαβ = 1

2(Wαβ −Wβα). (11)

This full coupling term is then

Sαβ = (ξAαν + Ωαν)(Qνβ + 13δνβ) + (Qαν + 1

3δαν)(ξAνβ − Ωνβ)

− 2ξ(Qαβ + 13δαβ)QννWνν . (12)

Here, ξ is a material-dependent that controls the relative importance of ro-tational and elongational flow for molecular alignment, and determines inpractice how the orientational order responds to a local shear flow. In allcases, the value used here is ξ = 0.7, which is within the ‘flow aligning’regime, where molecules align at a fixed angle (the Leslie angle) to the flowdirection in weak simple shear [40]. The value of the collective rotationaldiffusion used in all simulations is Γ = 0.5 in simulation units.

Hydrodynamics. The momentum evolution obeys a Navier-Stokes equa-tion driven by the divergence of a generalised stress Pαβ:

ρ ∂tuα + ρ uβ∂βuα = ∂βPαβ. (13)

The pressure tensor Pαβ is, in general, asymmetric and includes both viscousand thermodynamic components:

Pαβ = −p0δαβ + η∂αuβ + ∂βuα− ξHαγ

(Qγβ + 1

3δγβ)− ξ

(Qαγ + 1

3δαγ

)Hγβ

+ 2ξ(Qαβ + 1

3δαβ

)QγνHγν − ∂αQγν

δFδ∂βQγν

+ QαγHγβ −HαγQγβ. (14)

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A lattice Boltzmann (LB) flow solver is used, where the isotropic pressurep0 and viscous terms are managed directly by the solver (as in a simpleNewtonian fluid, of viscosity η), whereas the divergence of the remainingterms is treated as a local force density on the fluid. In all simulationsreported the mean fluid density ρ = 1 and the viscosity η = 0.01 (simulationsin confined geometry also used enhanced bulk viscosity ζ = 10η to ensurenumerical stability).

Surface boundary conditions for particles. Hydrodynamic boundaryconditions for solid objects are handled via the lattice Boltzmann solver. Inparticular, the standard method of bounce-back on links is used for bothparticles and confining walls [42, 43]. Boundary conditions for the orderparameter are set out in [45].

In all the simulations reported here the preferred orientation of the di-rector field at the particle surface is normal to the surface. The requirednematic director at the particle surface in a given location n0

α may then bedetermined from geometry alone, and a preferred order parameter Q0

αβ iscomputed via

Q0αβ = q0(n0

αn0β − 1

3δαβ). (15)

The magnitude of surface order q0 is set by

q0 = 23

(14

+ 34

√1 + (8/3γ)

)(16)

which corresponds to the bulk order in a defect-free nematically orderedsample [46] (this is also very close to the value of the order parameter in acholesteric or in a blue phase away from disclinations).

In all simulations the colloidal (hard sphere) radius is R = 2.3. To preventparticles overlapping at the hard sphere radius, an additional short range softpotential is included. This takes the form of V (h) = ε(σ/h) where h is thesurface to surface separation. The parameters are ε = 0.0004 and σ = 0.1 insimulation units. Both potential and resulting force are smoothly matchedto zero at a cut-off distance of hc = 0.25 simulation units.

Surface boundary conditions for walls. The confining solid walls, whichare flat and stationary, have the same normal boundary condition as at theparticle surface. In addition, degenerate planar anchoring is applied at thewalls, where the preferred nematic director may adopt any orientation par-allel to the surface. The preferred direction is determined explicitly by pro-

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jecting the local fluid order parameter to the plane of the wall [44]. Eq. 15and Eq. 16 are then used as before.

To prevent the particles being forced into the wall, a correction to thelubrication force on the colloidal particle is added for very low particle surfaceto wall surface separations h < 0.5 lattice spacing. The correction is basedon the analytic expression for the lubrication between a sphere and a flatsurface.

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AcknowledgementsWe acknowledge support by UK EPSRC grants EP/J007404/1 (KS and JSL),EP/G036136/1 (OH), and EP/I030298/1 (KS and JSL). MEC is supportedby the Royal Society. We thank both Hector and PRACE for computationalresources.

Author ContributionsKS, OH and JSL contributed to the code base for simulation and per-formed simulations. All authors helped to analyse the results and writethe manuscript.

Additional InformationCompeting financial interests: The authors declare no competing finan-cial interests.

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Figure 1: Bulk blue phase structures. Snapshots of the states obtainedafter dispersing a suspension of colloidal nanoparticles within a cholestericliquid crystal in the BPI-forming region. (a)–(d) correspond to the case of adispersion of particles in a pre-equilibrated BPI phase; (e)–(h) are obtainedby dispersing colloids in an isotropic phase and then quenching into therange where BPI is stable, leading to formation of an amorphous, BPIII-like,disclination network. Structures correspond to: (a) and (e) w = 0.23 andφ = 1%; (b) and (f) w = 0.23 and φ = 5%; (c) and (g) w = 2.3 andφ = 1%; (d) and (h) w = 2.3 and φ = 5%. The anchoring of the directorfield to the colloidal surface is normal. [For the full parameter list used togenerate Figs. 1-3 see Supplementary Notes.] For clarity, only a portion ofthe simulation box is shown its linear extent being ' 28R (one eighth of thetotal volume); the full structures are shown in Supplementary Figs. 2 and 3.

20

Figure 2: Confined blue phase structures. Snapshots of the steady statesobtained when a dispersion of colloids in the isotropic phase is placed in asandwich geometry, and then quenched into a regime where BPI is stable inthe bulk. Both top and side views are provided. The anchoring of the directorfield at the walls is normal for (a)–(d) and planar for (e)–(h). Structurescorrespond to: (a) and (e) w = 0.23 and φ = 1%; (b) and (f) w = 0.23and φ = 2%; (c) and (g) w = 2.3 and φ = 1%; (d) and (h) w = 2.3 andφ = 2%. As in Fig. 1, only one quarter of the simulation box is shown forclarity (its horizontal extent being ' 56R and vertical extent ' 24R in thenarrow direction); the full structures are shown in Supplementary Figs. 5and 6.

21

Figure 3: Structure in electric field. Steady states obtained when adispersion of colloids is inserted into a thermodynamically stable bulk BPIIIwith and without applied external electric field. Panel (a) shows the situationin zero field, and panel (c) the situation when the applied field is in the x-direction (perpendicular to the plane of the paper). Panels (b) and (d) showone plane of the three-dimensional structure factor S(k) computed from thepositions of the particles, with wavevector kx = 0, corresponding to (a)and (c) respectively. With field, the disclinations form a honeycomb patternwith hexagonal ordering perpendicular to the field direction; this is reflectedin the structure factor. Simulations use weakly anchoring particles w = 0.23,and a volume fraction of φ = 5% (see Supplementary Notes for full list ofparameters). The full simulation with a box size of ' 56R is shown.

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Supplementary Figures

Supplementary Figure 4: Blue phase structures. (A) Schematic represen-tation of the director field in a cholesteric (left) and within a double twistcylinder (right). (B) Snapshot of the disclination network in an equilibratedblue phase I structure. (C) Same as (B) for an amorphous blue phase IIInetwork.

Supplementary Figure 5: Bulk blue phase structures. As for mainmanuscript Fig. 1 a–d (with liquid crystal order parameter initialised toequilibrium blue phase I structure), but for the entire simulation system of1283 lattice sites. (A) solid volume fraction 1% and weak anchoring; (B) 4%solid volume fraction and weak anchoring; (C) 1% solid volume fraction andstrong anchoring; (D) 4% solid volume fraction and strong anchoring. Theview direction in the main figure is from the left here. There is a referenceparticle at the bottom left in each panel which does not take part in thesimulation.

Supplementary Figure 6: Bulk blue phase structures. As for mainmanuscript Fig. 1 e–h (initialised via a “quench” to generate a disorderednetwork), but for the entire simulation system of 1283 lattice sites. (E) solidvolume fraction 1% and weak anchoring; (F) 4% solid volume fraction andweak anchoring; (G) 1% solid volume fraction and strong anchoring; (H) 4%solid volume fraction and strong anchoring. Again, there is a reference parti-cle at the bottom left in each case which does not take part in the simulation.

Supplementary Figure 7: Free energy evolution. Plot of the free energy(per BP unit cell) versus time for the simulations corresponding to Fig. 1(a–h) in the main text.

Supplementary Figure 8: Confined blue phase structures. As for mainmanuscript Fig. 2 a–d (confined geometry with normal anchoring walls atboth top and bottom), but for the entire simulation system of 2562×56 latticesites. Each shows a top view and side view of the same simulation state. (A)solid volume fraction 1% and weak anchoring; (B) 4% solid volume fractionand weak anchoring; (C) 1% solid volume fraction and strong anchoring; (D)4% solid volume fraction and strong anchoring.

Supplementary Figure 9: Confined blue phase structures. As for mainmanuscript Fig. 2 e–h (confined geometry with planar anchoring walls atboth top and bottom), but for the entire simulation system of 2562×56 lat-tice sites. Each shows a top view and side view of the same simulation state.(E) colloid solid volume fraction 1% and weak anchoring; (F) 4% solid vol-ume fraction and weak anchoring; (G) 1% solid volume fraction and stronganchoring; (H) 4% solid volume fraction and strong anchoring.

Supplementary Figure 10: Structure with and without electric field.(Disclination lines are removed for clarity.) Panels A and B reproduce theresults in Fig. 3c and Fig. 3d respectively with the applied field in the x-direction (out of the plane of the paper). Also included is a cut of the struc-ture factor with ky = 0 (panel C). Panels D, E, and F show the correspondingsituation after the external field has been removed and the structure allowedto relax, and show residual hexagonal ordering. The structure factor datais for wave vectors kx = 0; (ky, kz) ∈ [−3π/8, 3π/8] (panels B and E) andky = 0; (kx, kz) ∈ [−3π/8, 3π/8] (panels C and F).

Supplementary Figure 11: Cycling the electric field. Starting from thefinal position of Supplementary Fig. 7 panel D, the field is re-applied, thistime in the y-direction. The corresponding view of the particle distributionis shown, again viewing along the field direction, in panel A. Correspondingstructure factor cuts with kx = 0 and ky = 0 are shown in panels B and C,respectively. The corresponding situation when the field is again switchedoff, and the structure allowed to relax, is shown in panels D, E, and F. Thewave vector values used are the same as in Supplementary Fig. 7.

Supplementary Figure 12: Cycling the electric field further. Startingfrom the final position of Supplementary Fig. 8 panel D, the field is re-appliedagain in the x−direction. Panel A shows the particle configuration viewedalong the field direction, and panels B and C show the structure factor withkx = 0 and ky = 0 respectively. The corresponding situation when the fieldis removed and the structure allowed to relax is shown in panels D, E, andF. The wave vector values are again as in Supplementary Fig. 7. Note thatthe on and off states are equivalent to those in Supplementary Fig. 7, upto a rotation. This is because all hexagonal structures perpendicular to thefield direction are energetically degenerate, and the actual orientation of thehexagon emerges spontaneously during switching.

Supplementary Table

A0 γ K p κ τ WR/K

Bulk BPI 0.01 3.086 0.007061 32√

2 0.6902 -0.2500 0.23, 2.3

Bulk Quench 0.01 3.086 0.007061 32√

2 0.6902 -0.2500 0.23, 2.3Confined 0.01 3.086 0.01897 64 0.8 -0.2500 0.23, 2.3External field 0.004 3.086 0.02 32 2.6 -0.2500 0.23

Supplementary Table 1: Free energy parameters used in the simula-tions. Bulk BPI correspond to Fig. 1 (a–d), bulk quench to Fig. 1 (e–h);confined geometries are shown in Fig. 2; and the external field case is relevantfor Fig. 3.

Supplementary Methods

Parameter details for bulk BPI. The parameters for the free energy arechosen to be representative of blue phase I: chirality κ ∼ 0.7 and reducedtemperature τ = −0.25. The full free energy parameters are shown in Ta-ble 1. The order parameter tensor Qαβ for bulk blue phase I is initialisedfrom an approximation in high chirality limit [1, 2]. Systems of 1283 latticesites are used with a pitch length of p = 32

√2, which accommodates 43

unit cells with fully periodic boundary conditions. Simulations at differentsolid volume fractions (1% and 4%), and different surface anchoring strengths(WR/K = 0.23 and WR/K = 2.3) representing “weak” and “strong” an-choring are run for two million simulation time steps. The initial distributionof the colloid positions is set at random for the required solid volume fraction;the colloids are initially at rest.

To generate a disordered network of disclination lines, a “quench” is per-formed in which the order parameter is initialised via a locally chosen ran-dom director, and Eq. 15 employed with a small amplitude q0 = 10−7 toprovide an order parameter. The mean-field spinodal point at γ = 3.0 isavoided and the parameters are the same as for the initial simulations withpitch p = 32

√2. The chirality and reduced temperature (see Table 1) remain

appropriate for BPI. Colloids are added as before and the simulations runfor 2 million simulation steps.

For simplicity, the size of the BPI unit cell is not allowed to readjust dynam-ically to minimise free energy (there is no “redshift” [1]): this would leadto quantitatively different values of the free energy, but qualitatively similarstructures.

Parameters for confined BPI. Here, a narrow sandwich of fluid is placedbetween flat walls in perpendicular to the narrow coordinate direction, andwith periodic boundaries in the other two directions. The system size usedin all cases is 2562×56 lattice sites. Each simulation is a “quench” in asimilar manner to that used in the bulk: the order parameter is initialisedrandomly. The free energy parameters (see Table 1) are again appropriatefor (equilibrium) blue phase I (chirality κ = 0.8 and reduced temperatureτ = −0.25). The cholesteric pitch is set to be p = 64, providing a slightlybetter resolution than the bulk simulations.

The surface anchoring for colloids is as before: always normal, but withWR/K = 0.23 and WR/K = 2.3. The colloid solid volume fractions are 1%and 4%. For both normal and planar anchoring at the walls, the strength isadjusted to correspond to that used for the strongly anchoring particles. Allsimulations are run for two million simulation time steps.

External electric field. For the simulations of BPIII with colloidal particlesa system size of 1283 is used with a cholesteric pitch of p = 32 in lattice units.The simulations are performed at chirality κ = 2.6 and reduced temperatureτ = −0.25. An initial configuration of randomly oriented and positioned

double twist cylinders is used, from which an amorphous BPIII networkemerges [3]. This structure is equilibrated for 6× 105 LB time steps until nosignificant further evolution is observed.

The colloidal particles with weak normal anchoring (WR/K = 0.23), arethen inserted with randomly chosen positions, and the composite system isequilibrated for another 1.2 × 106 time steps. At the end of the equilibra-tion phase the uniform electric field with reduced field strength E = 0.8 isswitched on (oriented along the z-direction), and the simulation run for a fur-ther 1.2×106 time steps. This is found to be sufficiently long for a hexagonalstructure to emerge and saturate. If the electric field is then switched off,and the system quickly relaxes to a metastable state which shows a resid-ual anisotropy in the colloidal distribution. The total length of this finalrelaxation phase is 1× 106 LB time steps.

Disclination rendering. In all cases visualisation of the defect lines iscarried out via an isosurface of the scalar order parameter q determined fromthe largest eigenvalue of the local order parameter tensor. A low value of thescalar order parameter q (typically in the range 0.12–0.14) unambiguouslyidentifies the disclinations.

Computation of the structure factor. To provide structural informationabout the field-aligned states and the residual anisotropy after switch off aprocedure similar to the one described in [3] is followed. The structure factorS(k) in the current approach is defined via the Fourier transform of thecolloid density F (k):

S(k) = |F (k)|2,with F (k) =∫d3rρ(r)eik.r. (17)

The colloid density ρ(r) is a discretised density and taken to be ρ(r) = 1 ifa lattice site at r is part of a colloid and ρ(r) = 0 otherwise. This definition,which deviates from the standard definition of a structure factor and doesnot consider the colloids as point-like objects, allows the same structuralinformation to be obtained in a simple way.

Parameter mapping to physical units. To get from simulation to physi-cal units, a calibration of scales of length, energy and time in the simulationsis required. For similar mappings see, e.g. [4].

The length scale can be set by mapping the half pitch to a realistic blue phaseunit cell size, which is in the 100–500 nm range [1]. For bulk simulations(Fig. 1 and Fig. 3 in the main text), a suitable choice is one where onesimulation unit (lattice site) corresponds to 10 nm. Therefore the colloidalsize in Figs. 1 and 3 corresponds to ∼ 50 nm.

To obtain an energy scale, it is possible to choose A0 ' 106 Pa, which isreasonable following Ref. [1] (this is also the choice in Ref. [5]). This choiceleads to a simulation unit of free energy (or stress) equal to 108 Pa in Figs. 1and 3. From the energy and length scales one can map elastic constants to

physical units: for instance, the simulation in Fig. 1 corresponds to a liquidcrystal with splay, bend and twist (Frank) elastic constants equal to 35 pN(or equivalently K = 70 pN).

The timescale calibration may be obtained from (see e.g. [6])

γ1 =2q2

Γ, (18)

which relates the rotational viscosity γ1 to the ordering strength q and theorder parameter mobility Γ . In the present simulations, Γ = 0.5 and thevalue of γ for the simulation in Fig. 1 leads to q = 1/2. For real liquidcrystalline materials, γ1 usually lies in range 10−2−1 Pa s [7]; for definitenesssay γ1 = 1 Pa s (equivalently 10 poise). Given the previous mapping for freeenergy (or stress) units, this leads to a simulation unit of time equal to 10−8 s.The total simulated time in Figs. 1-3 is therefore in the millisecond range.

To calibrate the electric field strength one can equate the value of the di-mensionless parameter E in the simulations with that of a hypothetical ex-periment. Given the previous value for A0, and a dielectric anisotropy of 20(or equivalently εa = 20ε0, with ε0 the dielectric permittivity of vacuum),an electric field of 100 V µm−1 is obtained. This corresponds to a value ofE ∼ 0.4 (considering a value of γ = 3.0857 as in Fig. 3 of the main text).

Given these mappings, all quantities can be easily transformed from simu-lation to physical units and vice versa. For example Figs. 1 and 2 use anisotropic viscosity η ' 0.01 and 0.1 in simulation units, which translate into0.01 − 0.1 Pa s respectively. This is sensible (if slightly low) for a molec-ular nematogen in the isotropic phase. The effective viscosity in orderedphases is higher, but this is ensured in our model by the coupling to theorder parameter. Note that the density is set to unity in the simulations:this corresponds to a fluid density which is about a thousand times largerthan in experiments. This makes no difference in practice as the Reynoldsnumber = ρV Λ/η (where V is a typical velocity, in our simulations around10−5 in lattice units, and Λ is a typical length scale, e.g. the size of a unitcell) remains small enough [8].

To summarise, the simulations represent BP-forming materials, with the in-terpretation of the simulation units for length, time, and energy densitybeing close to 10 nm, 10 ns, and 100 MPa respectively. For a Frank eleasticconstant of K = 70 pN and unit cell sizes in the range λ =100–500 nm, acorresponding shear modulus G ' K/λ2 = 0.3–7 kPa.

Supplementary References

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[3] Henrich, O., Stratford, K., Cates, M.E., and Marenduzzo, D., Structureof blue phase III of cholesteric liquid crystals, Phys. Rev. Lett. 106,107802 (2011).

[4] Denniston, C., Marenduzzo, D., Orlandini, E., and Yeomans, J.M., Lat-tice Boltzmann algorithm for three-dimensional liquid-crystal hydrody-namics, Philos. Trans. R. Soc. London, Ser. A 362, 1745 (2004).

[5] Henrich, O., Stratford, K., Marenduzzo, D., and Cates, M.E., Order-ing dynamics of blue phases entails kinetic stabilization of amorphousnetworks, Proc. Acad. Sci. USA 107, 1312 (2010).

[6] Denniston, C., Orlandini, E., and Yeomans, J.M., Lattice Boltzmannsimulattions of liquid crystal hydrodynamics, Phys. Rev. E 63, 056702(2001).

[7] de Gennes, P.-G., and Prost, J., The Physics of Liquid Crystals (Claren-don Press, 1995).

[8] Cates, M.E., et al., Simulating colloid hydrodynamics with lattice Boltz-mann, J. Phys. Condens. Matter 16, 3903 (2004).